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Consider the following weak version of the Noether–Enriques theorem (field is $\mathbb{C}$):

Let $\varphi:X\rightarrow Z$ be a morphism from a smooth projective surface onto a smooth curve with $F_z:=\varphi^{-1}(z)\cong\mathbb{P}^1$ for every point $z$. Then there exists a Zariski open neighborhood $U$ of any point $z$ making the following diagram commute. $\require{AMScd}$ \begin{CD} \varphi^{-1}(U) @>{\sim}>> U\times\mathbb{P}^1\\ @V{\varphi}VV @VV{\text{projection}}V\\ U @>{\sim}>> U \end{CD}

The critical point is to find a divisor $D$ on $X$ such that $D.F_z=1$ or to find a section $s:Z\rightarrow X$ for $\varphi$. Beauville states in his book Complex Algebraic Surfaces (c.f. Remark III.6) that this step can be done by using Tsen's lemma, and I want to know how exactly can we do this.

In particular, I want to know how to make $X$ a conic?

Indeed, Beauville even claims that we can use Tsen's lemma to show this step for the usual Noether–Enriques theorem.

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  • $\begingroup$ @LaurentMoret-Bailly Opps, it is a silly mistake. $\endgroup$
    – user485190
    Jul 3 at 14:12

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I will assume that $X$ is proper. Then the generic fiber is a smooth projective curve of genus $0$ over the function field $K = k(Z)$ but any such curve can be embedded as a conic in $\mathbb{P}^2_K$ using the anticanonical linear series. By Tsen's theorem, the conic has a $K$-point which we can spread out to a $U$-point of $X$ where $U$ is some open subset of $Z$. Finally by properness, this $U$-point extends to a section over $Z$.

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  • $\begingroup$ How can we make the $U$-point $U\rightarrow X_U$ to a section $Z\rightarrow X$? $\endgroup$
    – user485190
    Jul 4 at 10:30
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    $\begingroup$ Valuative criterion of properness. $\endgroup$
    – abx
    Jul 4 at 12:31

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